Abstract: We report the discovery of PSR J0952$-$0607, a 707-Hz binary millisecond
pulsar which is now the fastest-spinning neutron star known in the Galactic
field (i.e., outside of a globular cluster). PSR J0952$-$0607 was found using
LOFAR at a central observing frequency of 135 MHz, well below the 300 MHz to 3
GHz frequencies typically used in pulsar searches. The discovery is part of an
ongoing LOFAR survey targeting unassociated Fermi Large Area Telescope
$\gamma$-ray sources. PSR J0952$-$0607 is in a 6.42-hr orbit around a very
low-mass companion ($M_\mathrm{c}\gtrsim0.02$ M$_\odot$) and we identify a
strongly variable optical source, modulated at the orbital period of the
pulsar, as the binary companion. The light curve of the companion varies by 1.6
mag from $r^\prime=22.2$ at maximum to $r^\prime>23.8$, indicating that it is
irradiated by the pulsar wind. Swift observations place a 3-$\sigma$ upper
limit on the $0.3-10$ keV X-ray luminosity of $L_X < 1.1 \times 10^{31}$ erg
s$^{-1}$ (using the 0.97 kpc distance inferred from the dispersion measure).
Though no eclipses of the radio pulsar are observed, the properties of the
system classify it as a black widow binary. The radio pulsed spectrum of PSR
J0952$-$0607, as determined through flux density measurements at 150 and 350
MHz, is extremely steep with $\alpha\sim-3$ (where $S \propto \nu^{\alpha}$).
We discuss the growing evidence that the fastest-spinning radio pulsars have
exceptionally steep radio spectra, as well as the prospects for finding more
sources like PSR J0952$-$0607.